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On the move-to-front scheme with Markov dependent requests

Part of: Markov processes Theory of data

Published online by Cambridge University Press:  14 July 2016

R. M. Phatarfod ,
A. J. Pryde  and
David Dyte
R. M. Phatarfod*
Affiliation:
Monash University
A. J. Pryde*
Affiliation:
Monash University
David Dyte*
Affiliation:
Monash University
*
∗Postal address: Department of Mathematics, Monash University, Clayton, Victoria 3168, Australia.
∗Postal address: Department of Mathematics, Monash University, Clayton, Victoria 3168, Australia.
∗Postal address: Department of Mathematics, Monash University, Clayton, Victoria 3168, Australia.
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Abstract

In this paper we consider the operation of the move-to-front scheme where the requests form a Markov chain of N states with transition probability matrix P. It is shown that the configurations of items at successive requests form a Markov chain, and its transition probability matrix has eigenvalues that are the eigenvalues of all the principal submatrices of P except those of order N—1. We also show that the multiplicity of the eigenvalues of submatrices of order m is the number of derangements of N — m objects. The last result is shown to be true even if P is not a stochastic matrix.

Keywords

MOVE-TO-FRONTMARKOV CHAINTRANSITION MATRIXEIGENVALUES

MSC classification

Primary: 60J10: Markov chains (discrete-time Markov processes on discrete state spaces)
Secondary: 68P10: Searching and sorting 68P05: Data structures
Type
Short Communications
Information
Journal of Applied Probability , Volume 34 , Issue 3 , September 1997 , pp. 790 - 794
Copyright
Copyright © Applied Probability Trust 1997 

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